The universality of the Rezk nerve

Abstract

We functorially associate to each relative $\infty$-category $(R,W)$ a simplicial space $N_{\infty }^R(R,W)$, called its Rezk nerve (a straightforward generalization of Rezk's "classification diagram" construction for relative categories). We prove the following local and global universal properties of this construction: (i) that the complete Segal space generated by the Rezk nerve $N_{\infty }^R(R,W)$ is precisely the one corresponding to the localization $R[[W^{-1}]]$; and (ii) that the Rezk nerve functor defines an equivalence $RelCa{t}_{\infty }[[W_{BK}^{-1}]]\stackrel{\sim }{\to }Ca{t}_{\infty }$ from a localization of the $\infty$-category of relative $\infty$-categories to the $\infty$-category of $\infty$-categories.

References in corpus (1)

• From fractions to complete Segal spaces

Cited by in corpus (2)

• Algebras over not too little discs
• A Context for Manifold Calculus